Extended Theory

Calculations accuracy depends largely of size-interval introduzed and number of selected coefficients to calculate.
Use it is as follows.

1) Write the lower end of the range in the text box labeled Limit inf.
2) Enter the upper range in the text box labeled Limit Sup.
3) Write the function in the text box with the label function

In the piecewise function case, operate as follows
1) Write the lower end of the range in the text box labeled Limit inf.
2) Enter the upper range in the text box labeled Limit Sup.
3) Write the first function in the text box with the label function
.
4) Enter the upper sub-range in the text box labeled Subinterval 1.
5) Write the function as defined in the first sub-interval in the text box labeled subinterval 1
.

For example suppose we have the piecewise function
Then the fields are filled as

After the An, Bn calculations, is possible to plot the function and its Fourier Series by clicking "Show Graph". At this case

By default, the problem begins with the continuous function and the interval following
How it works?
To calculate Fourier coefficients integration methods seen in the numerical methods section are applied. With which we can approximate the integrals
In the case of Fourier coefficients, there are several methods to make the calculations, being exposed here created by the owners of Mathstools .
To calculate the derivative of the function: uses severeal numerical methods to dereivate.
To calculate the primitive function: numerical integration methods seen in the numerical methods section are applied.
Note that in numerical analysis, errors are obtained due to the particular methods and the limits of computer arithmetic.
In the the Fourier coefficients calculations case, it depends on the function and size of the chosen integration interval. In default probelm the error in calculating the Fourier coefficients is O (1e-8). For the numerical integration is O (1e-11) and in the derivative it is O (1e-14).